FileMaker is the 'standard' Mac Desktop Database package. They have a sort of 'lite' version called Bento.

No idea if that will help you - I don't think you can connect to an Access database from Filemaker -although I have never tried.

There are a couple of options for running Windows programs on a Mac though:
1) Virtualization - you can run a full copy of Windows in a virtual machine on your Mac. This can be tough on an Air because of limited drive space and limited RAM, although it you only have to run a few programs and then infrequently, you will probably be OK. There are many threads about this - do a search.

2) Remote Desktop - If you still have access to a windows desktop somewhere on your network, you can open a remote desktop connection to it. I personally prefer this to running a VM. MS makes a remote desktop client for the Mac - see Microsoft's Mac site -Mactopia.com

3) Wine - emulates Windows and allows some Windows programs to run natively without a VM. Pretty sure this won't work for Access anyway.